ORIENTATION EFFECTS IN BIPOLAR PLANETARY NEBULAE

The Astrophysical Journal, 675:380 388, 2008 March 1 # 2008. The American Astronomical Society. All rights reserved. Printed in U.S.A. ORIENTATION EFFECTS IN BIPOLAR PLANETARY NEBULAE Hugo E. Schwarz, 1 Hektor Monteiro, 1,2 and Ryan Peterson 3,4 Received 2006 September 8; accepted 2007 September 13 ABSTRACT We show that the inclination to the line of sight of bipolar planetary nebulae strongly affects some of their observed properties. We model these objects as having a dusty equatorial density enhancement that produces extinction that varies with the viewing angle. Our sample of 29 nebulae taken from the literature shows a clear correlation between the inclination angle and the near-infrared and optical photometric properties as well as the detected luminosity of the objects. As the inclination angle increases (the viewing angle is closer to the equatorial plane) the objects become redder, their apparent luminosity decreases, and their projected expansion velocity becomes smaller. We compute twodimensional models of stars embedded in dusty disklike structures of various shapes and compositions and show that the observed data can be reproduced by disk-star combinations with reasonable parameters. To compare with the observational data, we generate sets of model data by randomly varying the star and disklike structure parameters within a physically meaningful range. We conclude that a only a smooth pole to equator density gradient agrees with the observed phenomena. Subject headinggs: dust, extinction planetary nebulae: general radiative transfer 1. INTRODUCTION Planetary nebulae ( PNe) and symbiotic nebulae are the visible remains of a heavy mass-loss period near the end of the lives of most low- and intermediate-mass stars (Gurzadyan 1997). A fraction of asymmetrical PNe are bipolar ( BPNe), and this morphology is likely caused by binary central star systems (Soker 2004). BPNe have special properties (Corradi & Schwarz 1995) and are important objects for the study of outflows, mass loss, and binarity in PNe. Several meetings have been dedicated to asymmetrical PNe (Harpaz & Soker 1995; Kastner et al. 2000; Meixner et al. 2004, see references therein). The orientation angle on the plane of the sky of asymmetrical PNe was studied more than 30 years ago by Melnick & Harwit (1975) and more recently by Phillips (1997), who both found an apparent alignment of the long axes of PNe with the plane of the Galaxy. The paper by Corradi et al. (1998), based on a larger sample and using more rigorous statistical methods, showed that there is no significant alignment and that PNe are essentially randomly distributed on the sky. Assuming that this random distribution also holds for the inclination with respect to the line of sight, where the inclination i is taken to be 0 for a pole-on nebula and 90 for an object viewed with its main axis parallel to the plane of the sky, we investigate the possible effects of orientation on the observational properties PNe. Su et al. (2001) studied the observational orientation effects in six bipolar protoplanetary nebulae ( PPNe) and found that objects are redder when viewed edge-on, i.e., have the inclination angle i near 90, as expected from simple geometrical extinction considerations and assuming some equatorial density enhancement. From these data they concluded that PPNe generally have bipolar shapes and asymmetrical dust disks. Many works have studied dust content and modeled dust structures in astrophysical objects such as PPNe as well as PNe 1 Cerro Tololo Inter-American Observatory, Casilla 603, Colina El Pino S/N, La Serena, Chile. 2 Núcleo de Astrofísica Teórica-CETEC-UNICSUL, Rua Galvão Bueno, 868 São Paulo-SP CEP 01506-000, Brazil. 3 Lawrence University, Appleton, Wisconsin. 4 2003 CTIO/NOAO/AURA REU Program student. 380 as reviewed in Pascucci et al. (2004). In this work the authors also review and compare some of the most important codes available. Ueta & Meixner (2003) also developed a general purpose dust radiative transfer code to study axisymmetric systems (2-DUST). Most of the works done with these codes concentrated on modeling specific objects and their dust properties. These works tend to focus on PPNe and have not addressed large samples of PNe and the effects of orientation, in particular, in BPNe where there are still many questions as to weather dust can exist within the highly ionized regions and where extinction effects are expected to play a bigger role due to enhanced equatorial densities. Here we present a more general look at the phenomenon and apply a simple model to a larger sample of objects that we know to be bipolar, to see if we observe differences as a function of inclination angle. We study a range of overall geometries, dust properties, and different blackbody temperatures of central stars, and simulate observational data sets by letting these model parameters vary randomly over a restricted and physically meaningful range. We then compare these data sets directly with our observed results. 2. OBSERVATIONAL MATERIAL We have collected a sample of 29 BPNe and symbiotic nebulae that had sufficient data available in the literature to construct a reasonable spectral energy distribution (SED), with at least some data points in the optical, near-infrared ( NIR), and mid-infrared (MIR) wavelength ranges. Table 1 lists our sample of objects with the observational data compiled from the literature. After constructing the SEDs and obtaining the total flux we defined three different flux bands: visible (BVR), NIR (JHK ), and IRAS (12, 25, 60, and 100 m). We then obtained the relative flux in each band dividing these band fluxes by the total integrated flux. All fluxes were calculated by integrating the F(k) curve over the appropriate wavelength range. Possible differences in the line of sight interstellar extinctions were ignored. These differences wouldaffectmainlythebvr flux since the extinction is a strongly declining function of wavelength. Table 2 shows the calculated band fluxes, total fluxes (in Wm 2 ), and inclination angle for each nebula. The inclination

TABLE 1 Our Observed Sample of BPNe Name B V R J H K 12 (Jy) 25 (Jy) 60 (Jy) 100 (Jy) 19W32... 18.2 17.2 14.3 10.8 9.3 8.6 3.4 12.6 23.0 356.0 Hb 5... 10.3 8.2... 9.5 8.8 8.6 11.68 79.24 134.50 311.8 He 2-25... 16.4... 13.9 10.7 9.7 9.5 0.742 2.11 3.39 41.0 He 2-36... 11.9 11.2 8.9 9.8 9.6 9.4 0.42 4.88 6.40 8.8 He 2-111... 16.3 16.7... 13.5 11.1 9.7 1.28 3.08 11.21 117.2 He 2-114... 18.7... 16.3......... 2.0 0.38 3.08 38.46 He 2-145... 19.1... 16.4 12.0 10.9 10.1 3.1 2.67 44.75 201.0 IC 4406... 12.7... 11.4...... 10.1 0.33 2.69 21.08 25.35 07131 0147... 15.4... 11.7 10.2 9.5 9.1 2.59 4.22 3.96 3.68 K3-46... 19.1... 16.3 13.8 13.0 12.7 0.7 0.25 0.73 5.87 M1-8... 11.7... 10.4 10.7 10.5 9.8 0.25 0.66 3.10 5.54 M1-13... 11.0... 9.9......... 0.25 0.70 4.51 9.37 M1-16... 12.7... 10.0 13.0 13.2 12.2 0.32 2.33 9.45 7.59 M1-28... 18.9... 16.0 15.1 14.2 13.8 2.4 0.41 2.77 11.75 M1-91... 16.3 15.0 12.4......... 3.85 8.29 12.14 9.56 M2-9... 11.3... 9.5 10.9 9.2 7.0 50.50 110.20 123.60 75.84 M2-48... 15.2... 10.0 15.9 14.4 13.9 1.03 0.76 7.42 35.86 M3-28... 15.6... 10.3... 9.6 8.0 3.53 2.56 46.94 320.60 MyCn 18... 11.8... 9.6 11.8 10.3 9.8 1.80 20.66 24.28 13.24 Mz 1... 12.8... 11.6......... 0.48 1.40 14.45 92.96 Mz 3... 10.8... 8.0 9.2 7.4 5.6 88.76 343.20 277.00 112.60 NGC 650 51...... 17.7... 16.0 15.9 15.6 0.28 2.79 6.80 9.29 NGC 2346... 11.5 11.2... 10.2 9.4 8.4 0.47 0.88 7.97 13.40 NGC 2440... 11.8... 11.0 10.6 10.8 10.1 3.59 28.01 43.47 26.30 NGC 2818... 16.8... 16.8...... 18.2 0.32 1.00 2.30 2.89 NGC 2899... 17.8... 16.3......... 0.27 1.42 5.13 10.91 NGC 6072... 11.6... 10.1......... 0.38 2.87 24.89 31.17 NGC 6302...... 9.1 7.1 9.5 9.7 8.6 32.08 335.90 849.70 537.40 NGC 6445... 11.6... 12.5 10.9 9.6 9.2 1.50 15.01 44.44 43.23 NGC 6537... 11.0... 12.0 10.3 10.4 9.3 7.72 58.30 189.90 166.10 NGC 7026... 15.3 14.2 11.7 10.2 11.0 9.6 2.39 18.35 42.74 30.91 Sa 2-237... 16.1 15.5 15.0 13.1 12.6 11.8 1.17 6.01 16.56 8.29 He 2-104... 13.5 14.6 11.1 10.8 8.9 7.0 8.56 9.09 6.83 9.77 BI Cru... 12.5 10.9 11.3 7.3 6.0 4.8 17.29 15.34 11.84 119.10 R Aqr... 10.0 6.2... 0.5 0.1 0.8 1577.0 543.80 66.65 16.60

382 SCHWARZ, MONTEIRO, & PETERSON Vol. 675 TABLE 2 Integrated BVR, JHK, and IRAS and Total Fluxes as well as Inclinations for Objects Studied Name BVR JHK IRAS F Total (W m 2 ) Incl. 19W 32... 2.82479E 14 8.95469E 13 1.41920E 11 1.51157E 11 80 Hb 5... 2.39058E 11 1.54571E 12 2.85078E 11 5.39593E 11 65 He 2-25... 6.04131E 14 6.10470E 13 1.83820E 12 2.50908E 12 90 He 2-36... 4.65515E 12 9.10518E 13 1.27460E 12 6.84027E 12 30 He 2-111... 1.91574E 14 2.07320E 13 4.76610E 12 4.99258E 12 70 He 2-114... 6.76778E 15 4.52091E 14 1.85340E 12 1.90538E 12 45 He 2-145... 5.81293E 15 2.31164E 13 9.36290E 12 9.59988E 12 55 IC 4406... 8.78346E 13 8.53294E 13 2.21980E 12 3.95144E 12 65 07131 0147... 3.89401E 13 8.45939E 13 1.46230E 12 2.69764E 12 60 K3-46... 6.26481E 15 3.18332E 14 4.17600E 13 4.55698E 13 65 M1-8... 2.20631E 12 4.37448E 13 4.62900E 13 3.10665E 12 25 M1-13... 3.82688E 12 9.05111E 12 6.53100E 13 1.35311E 11 25 M1-16... 2.11055E 12 4.67345E 14 1.05980E 12 3.21708E 12 60 M1-28... 8.12765E 15 1.04420E 14 1.14020E 12 1.15877E 12 55 M1-91... 1.67529E 13 9.43910E 13 2.85110E 12 3.96254E 12 75 M2-9... 4.18374E 12 2.01761E 12 3.43042E 11 4.05055E 11 75 M2-48... 1.74157E 12 7.66465E 15 1.79550E 12 3.54474E 12 45 M3-28... 1.31838E 12 1.66314E 12 1.31547E 11 1.61362E 11 60 MyCn 18... 3.39726E 12 3.34574E 13 4.54040E 12 8.27224E 12 55 Mz 1... 7.63475E 13 1.89538E 12 3.79930E 12 6.45816E 12 50 Mz 3... 1.30890E 11 8.20062E 12 8.06020E 11 1.01892E 10 65 NGC 650 51... 5.09530E 15 2.93012E 15 1.02350E 12 1.03153E 12 60 NGC 2346... 2.12800E 12 1.06373E 12 1.02360E 12 4.21533E 12 45 NGC 2440... 1.61619E 12 4.00061E 13 7.22120E 12 9.23745E 12 65 NGC 2818... 1.26322E 14 5.05320E 15 4.01700E 13 4.19385E 13 90 NGC 2899... 8.87462E 15 2.76607E 14 8.21700E 13 8.58235E 13 75 NGC 6072... 2.68009E 12 7.53013E 12 2.61900E 12 1.28292E 11 45 NGC 6302... 2.06389E 11 1.20271E 12 1.06935E 10 1.28777E 10 70 NGC 6445... 1.29920E 12 6.47602E 13 5.69510E 12 7.64190E 12 50 NGC 6537... 2.23175E 12 6.03036E 13 2.34040E 11 2.62388E 11 60 NGC 7026... 3.28287E 13 5.30525E 13 5.86380E 12 6.72261E 12 50 Sa 2-237... 3.60786E 14 5.76959E 14 2.09040E 12 2.18417E 12 70 He 2-104... 6.02155E 13 2.16783E 12 3.86540E 12 6.63539E 12 50 BI Cru... 1.56904E 12 2.34863E 11 1.03283E 11 3.53837E 11 40 R Aqr... 4.30754E 10 6.52070E 09 4.63337E 10 7.41479E 09 20 Note. See text for the detailed definition of these quantities. Fig. 1. Observed relative fluxes in the BVR (open triangles), JHK (gray dots), and IRAS (black squares) bands as a function of the inclination angle. angle is an average of the three estimated inclination angles for each object. The inclinations are determined from visual inspection of narrowband images available in the literature. We assume that the equatorial cross section of all spatial structures is circular and assign an inclination angle based on the apparent shape of the bright central areas of the nebulae in their respective narrowband images. In cases in which no central bright structure was present or resolved, we used the full structural appearance to assign an angle. These are clearly the objects that present the greatest uncertainties in the inclination angle determined. The inclination angles were determined in this manner by the three authors independently, to obtain an idea of the associated uncertainties in deriving inclinations from optical images. The process has a subjective component, but we found (somewhat to our surprise) that for nearly all cases our independent estimates were in agreement to within 10 or so, accurate enough for the purposes of this paper. For only three recalcitrant objects did we differ by as many as 30 in our estimates. Typical standard deviations of the mean of the three estimates are 5 8, and in all cases they were below 18. In Figure 1 we plot the relative fluxes for each band (visible [BVR], NIR [JHK], and MIR [IRAS bands]) as a function of the

No. 1, 2008 ORIENTATION EFFECTS IN BIPOLAR NEBULAE 383 Fig. 2. Benchmark results obtained with our simplified code. The models in this graph have the same characteristics as those calculated by Pascucci et al. (2004). This figure should be compared to Fig. 7 of Pascucci et al. (2004). The optical depths () indicated are calculated for v ¼ 0:55 m. inclination angle. One can clearly see the increase of relative MIR as well as the decrease of JHK and BVR with increasing inclination for the sample. 3. MODELS 3.1. A Simplified Two-dimensional Radiative Transfer Code To try to explain this observed behavior, we propose that these bipolar systems are composed of a dusty equatorial density enhancement being irradiated by a star or a binary system. The ultraviolet (UV) flux from this system heats up the dust that then re-emits the radiation in the IR. The combined effect of this reradiation with the extinction produced by the density distribution is what produces the observed effect. To simulate this system we created a simple two-dimensional radiative transfer code using the IDL package. This code works by creating a two-dimensional grid of cells, each having a given density. By requiring conservation of luminosity at each cell and iterating, we determine the temperature of the dust. The temperature is determined from the equation Z Z C abs cu k dk ¼ C abs 4B k (T) dk; ð1þ where T is the temperature, u k is the energy density of the radiation field, and C abs is the absorption cross section, shown in Figure 2 as a function of the wavelength. Scattering is assumed to happen only once (single-scattering) and is computed assuming an isotropic phase function, therefore not being a strict calculation. Some other simplifying assumptions are made concerning the dust grains: 1. The dust species are well mixed, allowing for the use of average cross sections. 2. The dust mixture is in equilibrium with the radiation field which implies a single dust temperature for all dust species. 3. The dust-to-gas ratio is the same as that for the interstellar medium. 4. The dust is mixed with the gas throughout the structure. 5. The central star (or stars) is considered a point source and is located at the center of the grid. In cases in which multiple stars were considered, their combined spectra were placed at the central point of the grid. Our simplified code also assumes that the grid points are optically thin to radiation emitted by other grid points. This combined with the assumption of single scattering allows us to map the two-dimensional grids to three-dimensional cubes. With the twodimensional temperature and density grids, we constructed a threedimensional cube where the calculated intensity emitted by the dust plus scattered radiation (assumed isotropic) is obtained for each cell. Using this cube and the extinction cross section from Li & Draine (2001), we calculated the image projected onto the sky for a given line of sight. The total emitted radiation by the structure was obtained from this projected image and could then be compared to observational results after calculating the relative luminosities in each band, noting that these relative quantities are distance independent. Fig. 3. Absorption cross sections used in the model calculations. From Li &Draine(2001).

Fig. 4. Density and temperature maps obtained for disklike structure as well as their respective relative luminosity relation plot with BVR luminosity band represented by open triangles, JHK by open circles, and IRAS by filled squares. All disklike structures have 55nm ¼ 10 in the equator.

ORIENTATION EFFECTS IN BIPOLAR NEBULAE 385 TABLE 3 Model Parameters from Pascucci et al. (2004) Symbol Parameter Value M *... Stellar mass 1 M R *... Stellar radius 1 R T *... Stellar effective temperature 5800 K R out... Outer radius 1000 AU R in... Inner radius 1 AU z d... Disk height 125 AU a... Grain radius 0.12 m g... Grain density 3.6 g cm 3 v... Optical depth at 550 nm 0.1, 1, 10, 100 3.2. Benchmark Results To check the consistency of the results obtained from this simplified code, we compared our results to those obtained by Pascucci et al. (2004). In that work the authors test five different codes with a well defined set of two-dimensional models having distinct visual optical depths. All models were composed of a stellar central source embedded in a disk structure made up of spherical silicate dust grains with a radius of 0.12 m and a density of 3.6 g cm 3. The SEDs obtained are presented for two distinct inclination angles and four different optical depths. For the details of disk configuration and dust characteristics refer to Pascucci et al. (2004). We calculated models for the same disk and dust configuration studied by Pascucci et al. (2004). In Figures 3 and 4 we present our results. These figures should be compared with Figure 7 of Pascucci et al. (2004). The main characteristics of the models are presented in Table 3. 3.3. Calculated Models For the dust characteristics in the orientation study, we adopted typical interstellar grains with R v ¼ 3:1 and absorption cross section calculated by Li & Draine (2001) with a dust-to-gas ratio of 0.008. The absorption cross section is shown in Figure 3. Using the simple code described above, we initially calculated models for the three basic disklike structures shown in the left panels of Figure 4. From top to bottom these are (a) a torus, (b)a flat disk, and (c) a curved disk. A fourth distribution (d) was used with a much more smoothly varying density gradient from pole to equator, shown as the bottom left panel Figure 4. For simplicity we adopted the same physical size and peak density for all structures. The structures were constructed such that 55nm ¼ 10 through the equator in all cases. For the four disk shapes studied we obtained the temperature maps shown in the middle panels of Figure 4. These maps show the shadowing effect of the dense structures. With the calculated SEDs, we also constructed plots with the correlations of the relative luminosities for each of the structures as a function of inclination angle, also shown in the right panels of Figure 4. These can then be compared to the plots of Figure 1, which shows the observed behavior. It is not trivial to compare the model result with the much noisier observational plot. The noise, or spread, is likely caused by stellar and dust properties varying from object to object and also by the errors on estimating the inclination angles for each individual nebula. To make the comparison easier, we generated sets of model data by allowing the stellar and dust properties to vary randomly within a physically meaningful range of values. This simulates observing a large sample of real objects with difference parameters as was done in the observed sample, allowing a direct comparison of the model and observational samples. To accomplish this we ran a set of 29 models, equivalent to the observational sample we had, which are plotted in Figure 5. For each object the code was executed with input parameters selected in a random fashion with boundaries as described in Table 4. The disklike structure was obtained by using the same density equation given by Pascucci et al. (2004) with parameters also listed in Table 4. The parameters presented in Table 4 were allowed to vary in a uniform random distribution in the given interval. The average total mass in the generated structures is 0.005 M. 4. LUMINOSITY EFFECTS Another predicted effect is that high-inclination objects should have lower detected luminosities, given a central ionizing source, because only the equatorial donut is seen, while for lowinclination objects the central object and donut are observed. Clearly, this effect is only possible in objects with an asymmetrical matter distribution where extinction varies with the line of sight. To check the predicted behavior, we generated a sample in the same way done for the relative luminosity effect discussed above and plotted the detected luminosity as a function of inclination angle. The plot is shown in Figure 6. Our results are also consistent with those obtained by Whitney et al. (2003), where the authors conduct similar modeling of various dust structures to investigate effects of geometry in protostellar envelopes using two-dimensional radiative transfer code. In particular they also show that the orientation can influence the observed SED as well as presenting correction factors for luminosities obtained from them. For a subset of seven objects, we have reliable distances and therefore were able to determine their absolute luminosities. These objects are listed in Table 5 together with their luminosities, distances, and inclination angles. The average luminosities for objects in Table 5 are (1) with i 45 ; L ¼ 346 L and (2) for i 45 ; L ¼ 3550 L. There is a significant difference in the mean luminosities of the two groups, whereby the high-i group has the lower luminosity, as predicted. The standard deviation of the high-i group mean is 154 and of the low-i group 1060. The difference between the mean values is 3204 nearly a 3 result. A check on the randomness of the angle distribution on the sky is to count the number of objects in inclination angle bins and compare these with the theoretically expected numbers which should go as sin (i) if the distribution on the sky is truly random. We have 7% of objects in the 0 30 bin, 52% in the 31 60 bin, and 41% in the 61 90 bin. Noting that there are five objects with 60 inclinations which happen to fall in the 31 60 bin. Distributing these equally over this and the next bin, we obtain 7%, 43%, and 50%. Theory predicts, by integrating sin (i) over the same three angle bins, respectively, 13%, 37%, and 50% of the objects in the bins, relatively close to the observed values, with or without the 60 object correction. 5. PROJECTED EXPANSION VELOCITIES The expansion velocities of BPNe with low inclinations should, all other things being equal, be on average higher than those of high-inclination objects. To test this idea for our sample, we used the published expansion velocities from Corradi & Schwarz (1995). To correct as much as possible for intrinsic differences in expansion velocities between objects, we took the aspect ratio of an object to be proportional to its expansion velocity. This

Fig. 5. Model-generated data sets for comparison with observed data. The parameters are allowed to vary randomly over a range defined in the text. The five graphs respectively show all parameters varying, only the disk size varying, the maximum density, the giant companion, and the white dwarf, all as indicated in the figure. 386

TABLE 4 Model Parameter Intervals for Random Sample Symbol Interval T wd... (0.5 1) ; 10 5 K L wd... (2 5) ; 10 2 L T giant... (3 6) ; 10 3 K L giant... (0.5 1) ; 10 3 L R out... (0.55 1) ; 10 3 AU R in... 10 AU a... 0.001 1.0 m g... 3.6 g cm 3 v... 15 20 TABLE 5 Objects for which Distances are Known with Their Observed Luminosities Object L d (L (pc) i (deg) Sa 2-237... 340 2100 70 M2-9... 553 640 75 He 2-104... 205 800 50 He 2-111... 440 2800 70 M1-16... 194 1800 70 R Aqr... 2800 200 20 BI Cru... 4300 180 40 Fig. 6. Plot of the detected luminosities of a sample of model BPNe as a function of inclination angle. Fig. 7. Plot of the expansion velocities of our sample as a function of inclination angle corrected for the objects aspect ratio. 387

388 SCHWARZ, MONTEIRO, & PETERSON TABLE 6 Objects for which the Expansion Velocity is Known with Their Aspect Ratio Object V exp (km s 1 ) Aspect Ratio (AR) 19W32... 9 5.0 Hb 5... 106 2.2 He 2-36... 70 1.9 He 2-111... 127 4.0 IC 4406... 25 3.3 M1-16... 127 5.3 M1-91... 4 5.0 M2-9... 69 12.0 MyCn 18... 14 2.0 Mz 3... 76 3.8 NGC 2346... 35 2.5 NGC 2440... 32 2.0 NGC 2899... 13 1.9 NGC 6302... 55 3.5 NGC 6445... 42 1.5 NGC 6537... 150 4.0 NGC 7026... 37 1.8 He 2-104... 125 10.0 BI Cru... 214 10.0 makes sense because the polar expansion compared to the typical PNe expansion of 15 km s 1 determines to some order the aspect ratio of the object. Plotting the expansion velocity divided by aspect ratio against the inclination angle for all those objects for which we have data, we see in Figure 7 that there is a correlation between these parameters. When plotting the expansion velocity without correcting for the aspect ratio, the correlation is also there. This is unlikely to be physical and must be related to the projection of the true expansion velocities. In Table 6 we list the objects with their expansion velocities and aspect ratios. 6. CONCLUSIONS In this work we investigate the orientation effects present in BPNe. The data available in the literature clearly shows the correlation between the inclination angle and the IR, near-ir, and optical photometric properties as well as the detected luminosity and expansion velocity of the objects. Our simplified dust model, designed to study the observed correlations, copes well with typical density structures and ionizing central sources as shown by the benchmark results presented in x 3. The most important result of this work as discussed previously is that a sharply changing density distribution cannot reproduce the observed behavior of the relative luminosity with inclination angle. The only models that agree reasonably well with the observations have the smoothly varying structure of the bottom row of Figure 4. This structure is not a disk in the strict sense of the word but presents a denser equatorial region that resembles one. Exploring the parameter space with the models, we found that using a hot, compact star with a cooler giantlike companion, we get good agreement with the observations. The presence of the cooler companion, however, is not necessary to reproduce the observations as seen by the small variations introduced by the presence of a random secondary with typical giant parameters. Since we have only a relatively small sample of random systems, the spread in our plots is rather large and a precise and detailed model cannot be selected on this basis. We do, however, confirm that the observed behavior as a function of inclination angle is easily reproducible with simple and physically meaningful and plausible models of bipolar nebulae. We should note that there are other processes present in the objects studied in this work, in particular photoionization ( being the most important), producing the narrow emission lines which are widely studied. These lines should indeed introduce some contamination in the BVR measurements of the central sources beingthatitisverydifficulttoextractthecontributionofthe nebula perfectly. However, this difference will not be relevant for the general result studied in this work, since only a small portion of the total matter present is ionized, making the contribution on the final SED small. Some issues are still open, as the gas-dust mixing question and precise disk shape, among others, cannot be addressed properly with the current data set and would require better imaging and photometry as well as more developed three-dimensional modeling to be studied precisely. However, the existence of a structure with high equatorial matter concentration, likely to be one of the important reasons for the shaping of BPNe, is clearly demonstrated by our data set and models. H. Monteiro would like to acknowledge the support of FAPESP through grant 2007/00956-6 and the suggestions of the referee which greatly improved the work. Dr. Hugo Schwarz was deceased, 2006 October 20. An obituary will be published in an upcoming issue of the Bulletin of the American Astronomical Society, and other personal recollections can be found at http://www.subjectivelens.com/ Hugo/. Ryan Peterson is grateful for the opportunity to participate in the Research Experience for Undergraduates ( REU) Program of the National Science Foundation. Cerro Tololo Interamerican Observatory, National Optical Astronomy Observatory, operated by the Association of Universities for Research in Astronomy, Inc., under a cooperative agreement with the National Science Foundation. Corradi, R. L. M., Aznar, R., & Mampaso, A. 1998, MNRAS, 297, 617 Corradi, R. L. M., & Schwarz, H. E. 1995, A&A, 293, 871 Gurzadyan, G. A. 1997, The Physics and Dynamics of Planetary Nebulae (Berlin: Springer) Harpaz, A., & Soker, N., eds. 1995, Asymmetrical Planetary Nebulae, Annals of the Israel Physical Society, 11 Kastner, J. H., Soker, N., & Rappaport, S. A., eds. 2000, ASP Conf. Proc. 199, Asymmetrical Planetary Nebulae II (San Francisco: ASP) Li, A., & Draine, B. T. 2001, ApJ, 554, 778 Melnick, G., & Harwit, M. 1975, MNRAS, 171, 441 REFERENCES Meixner, M., Kastner, J. H., Balick, B., & Soker, N., eds. 2004, ASP Conf. Proc. 313, Asymmetrical Planetary Nebulae III (San Francisco: ASP) Pascucci, I., Wolf, S., Steinacker, J., Dullemond, C. P., Henning, T., Niccolini, G., Woitke, P., & Lopez, B. 2004, A&A, 417, 793 Phillips, J. P. 1997, A&A, 325, 755 Soker, N. 2004, in ASP Conf. Proc. 313, Asymmetrical Planetary Nebulae III, ed. M. Meixner et al. (San Francisco: ASP), 562 Su, K. Y. L., Hrivnak, B. J., & Kwok, S. 2001, AJ, 122, 1525 Ueta, T., & Meixner, M. 2003, ApJ, 586, 1338 Whitney, B. A., Wood, K., Bjorkman, J. E., & Wolff, M. J. 2003, ApJ, 591, 1049